Sally75
It's unfortunate that Robert Dent seems to have died at an awkward time! It isn't necessary to know the exact date of his death, the year is near enough. The grant of probate usually specifies the date and place of death anyway.
Probate of Robert Dent's Will would probably have been granted within 6 months or a year of his death. Until early in 1858 Probate would have been granted by the relevant Ecclesiastical (Consistory) Court - almost certainly Durham in this case. [I believe the National Archive website only covers PCC Wills - i.e. probates granted by the Prerogative Court of Canterbury].
From 1858 the granting of Probate was transferred to the State and an annual alphabetical index is kept of all Grants of Probate and Administration. You will find a very informative article on the website of the Northumberland and Durham Family History Society at
http://www.ndfhs.org.uk/Articles/Wills/Wills1.htmlThere's also an interesting item at
http://www.dur.ac.uk/news/newsitem/?itemno=4535&rehref=%2Fnews%2Farchive%2F&resubj=%20HeadlinesI have been fortunate enough to find that one of my local Record Offices has copies of all the Probate Indexes since 1858, so it might not be necessary to get to the Principal Probate Registry or apply to York, if you are in the UK.
Teea